Research Article Gregus-Type Common Fixed Point Theorems for Tangential Multivalued Mappings of Integral Type in Metric Spaces

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1 International Mathematics and Mathematical Sciences Volume 211, Article ID , 12 pages doi:1.1155/211/ Research Article Gregus-Type Common Fixed Point Theorems for Tangential Multivalued Mappings of Integral Type in Metric Spaces W. Sintunavarat and P. Kumam Department of Mathematics, Faculty of Science, King Mongkut s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 114, Thailand Correspondence should be addressed to P. Kumam, poom.kum@kmutt.ac.th Received 28 November 21; Accepted 11 February 211 Academic Editor: Frank Werner Copyright q 211 W. Sintunavarat and P. Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The concept of tangential for single-valued mappings is extended to multivalued mappings and used to prove the existence of a common fixed point theorem of Gregus type for four mappings satisfying a strict general contractive condition of integral type. Consequently, several known fixed point results generalized and improved the corresponding recent result of Pathak and Shahzad 29 and many authors. 1. Introduction The first important result on fixed points for contractive-type mappings was the well-known Banach contraction principle, published for the first time in 1922 in 1 see also 2. Banach contraction principle has been extended in many different directions, see 3 5, andso forth. Many authors in 3, 5 12 established fixed point theorems involving more general contractive conditions. In 1969, Nadler 13 combines the ideas of set-valued mapping and Lipschitz mapping and prove some fixed point theorems about multivalued contraction mappings. Afterward, the study of fixed points for multivalued contractions using the Hausdorff metric was initiated by Markin 14. Later, an interesting and rich fixed point theory for such maps was developed see The theory of multivalued maps has applications in optimization problems, control theory, differential equations, and economics. Sessa 19 introduced the concept of weakly commuting maps. Jungck 2 defined the notion of compatible maps in order to generalize the concept of weak commutativity and showed that weakly commuting mappings are compatible but the converse is not true.

2 2 International Mathematics and Mathematical Sciences This concept was further improved by Jungck and Rhoades 21 with the notion of weakly compatible mappings. In 22, Aamri and Moutawakil 22 defined property E.A. This concept was frequently used to prove existence theorems in common fixed point theory. Three years later, Liu et al. 23 introduced common property E.A. Theclassof E.A maps contains the class of noncompatible maps. Branciari 3 studied contractive conditions of integral type, giving an integral version of the Banach contraction principle, that could be extended to more general contractive conditions. Recently, Pathak and Shahzad 24 introduced the new concept of weak tangent point and tangential property for single-valued mappings and established common fixed point theorems. Very recently, Vetro 25 obtained an interesting theorem for mappings satisfying a contractive condition of integral type which is a generalization of Branciari 3, Theorem2. The aim of this paper is to define a tangential property for multivalued mappings which generalize the concept of tangential property for single-valued mappings of Pathak and Shahzad 24 and prove a common fixed point theorem of Gregus type for four mappings satisfying a strict general contractive condition of integral type. 2. Preinary Throughout this paper, X, d denotes a metric space. We denote by CB X, theclassofall nonempty bounded closed subsets of X. TheHausdorff metric induced by d on CB X is given by H A, B max { sup a A d a, B, sup d b, A b B }, 2.1 for every A, B CB X, whered a, B d B, a inf{d a, b : b B} is the distance from a to B X. Letf : X X and T : X CB X. Apointx X is a fixed point of f resp. T if fx x resp. x Tx. The set of all fixed points of f resp. T is denoted by F f resp. F T. Apointx X is a coincidence point of f and T if fx Tx.Thesetofall coincidence points of f and T is denoted by C f, T. Apointx X is a common fixed point of f and T if x fx Tx. The set of all common fixed points of f and T is denoted by F f, T. Definition 2.1. The maps f : X X and g : X X are said to be commuting if fgx gfx, for all x X. Definition 2.2 see 19. Themapsf : X X and g : X X are said to be weakly commuting if d fgx,gfx d fx,gx, forallx X. Definition 2.3 see 2. Themapsf : X X and g : X X are said to be compatible if d fgx n,gfx n whenever {x n } is a sequence in X such that fx n gx n z, forsomez X. Definition 2.4 see 26. Themapsf : X X and g : X X are said to be weakly compatible fgx gfx,forallx C f, g.

3 International Mathematics and Mathematical Sciences 3 Definition 2.5 see 22. Letf : X X and g : X X. Thepair f, g satisfies property E.A if there exist the sequence {x n } in X such that fx n gx n z X. 2.2 See example of property E.A in Kamran 27, 28 and Sintunavarat and Kumam 11. Definition 2.6 see 23. Letf, g, A, B : X X. Thepair f, g and A, B satisfy a common property E.A if there exist sequences {x n } and {y n } in X such that fx n gx n Ay n By n z X. 2.3 Remark 2.7. If A f, B g, and{x n } {y n } in 2.3, then we get the definition of property E.A. Definition 2.8 see 24. Letf, g : X X. Apointz X is said to be a weak tangent point to f, g if there exist sequences {x n } and {y n } in X such that fx n gy n z X. 2.4 Remark 2.9. If {x n } {y n } in 2.4, we get the definition of property E.A. Definition 2.1 see 24. Letf, g, A, B : X X. Thepair f, g is called tangential with respect to the pair A, B if there exist sequences {x n } and {y n } in X such that fx n gy n Ax n By n z X Main Results In this section, we first introduce the notion of tangential property for two single-valued and two multivalued mappings. Throughout this section, R denotes the set of nonnegative real numbers. Definition 3.1. Let f, g : X X and A, B : X CB X. Thepair f, g is called tangential with respect to the pair A, B if Ax n By n D CB X, 3.1 whenever sequences {x n } and {y n } in X such that fx n gy n z D, 3.2 for some z X.

4 4 International Mathematics and Mathematical Sciences Example 3.2. Let R,d be a metric space with usual metric d. Letf, g : R R and A, B : R CB R be mappings defined by fx x 1, gx x 2, Ax {x 2 /2 1}, and Bx {x 2 2}, forallx R. Clearly, there exists two sequences {x n 2 1/n} and {y n 1 1/n} such that Ax n By n {3} CB R 3.3 whenever fx n gy n 3 R. 3.4 So, the pair f, g is tangential with respect to the pair A, B. Definition 3.3. Let f : X X and A : X CB X. The mapping f is called tangential with respect to the mapping A if Ax n Ay n D CB X, 3.5 whenever sequences {x n } and {y n } in X such that fx n fy n z D, 3.6 for some z X. Example 3.4. Let R,d be a metric space with usual metric d.letf : R R and A : R CB R be mappings defined by fx x 1, Ax { } x Clearly, there exist two sequences {x n 1 1/n} and {y n 1 1/n} such that Ax n Ay n {2} CB R 3.8 whenever fx n fy n 2 R. 3.9 So, the mapping f is tangential with respect to the mapping A. Now,westateandproveourmainresult.

5 International Mathematics and Mathematical Sciences 5 Theorem 3.5. Let f, g : X X and A, B : X CB X satisfy ( 1 α <α d fx,gy ) H Ax,By ( d Ax,fx d By,gy a d fx,gy 1 a max d Ax,gy d Ax,fx, d fx,by d By,gy, ) ( d Ax,fx ) 1/2 ( d Ax,gy 1/2 ), ( d fx,by ) 1/2 ( d Ax,gy ) 1/2, 3.1 for all x, y X for which the right-hand side of 3.1 is positive, where <a<1, α and ψ : R R is a Lebesgue integrable mapping which is a summable nonnegative and such that ɛ >, 3.11 for each ɛ>. If the following conditions (a) (d) hold: a there exists a point z f X g X which is a weak tangent point to f, g, b f, g is tangential with respect to A, B, c ffa fa, ggb gb,andafa Bgb for a C f, A and b C g,b, d the pairs f, A and g,b are weakly compatible. Then, f, g, A,andB have a common fixed point in X. Proof. Since z f X g X, z fu gv for some u, v X. It follows from a point z which is a weak tangent point to f, g that there exist sequences {x n } and {y n } in X such that fx n gy n z. 3.12

6 6 International Mathematics and Mathematical Sciences Because the pair f, g is tangential with respect to the pair A, B, weget Ax n By n D, 3.13 for some D CB X. Sincez fu gv and 3.12 and 3.13 are true, we have z fu gv fx n gy n Ax n By n D ( 1 α We claim that z Bv. If not, then condition 3.1 implies d fxn,gv ) H Axn,Bv ( d Axn,fx n <α a d Bv,gv d Axn,gv d fxn,gv 1 a max d Axn,fx n, d fxn,bv d Bv,gv, ) ( d Axn ),fx n 1/2 ( d Axn,gv 1/2 ), ( d fxn,bv ) 1/2 ( d Axn ),gv 1/ Letting,weget H D,Bv 1 a d Bv,z Since d z,bv < H D,Bv 1 a d Bv,z 3.17 < d z,bv, which is a contradiction, then z Bv.

7 International Mathematics and Mathematical Sciences 7 Again, we claim that z Au. If not, then condition 3.1 implies ( 1 α <α d fu,gyn ) H Au,Byn ( d Au,fu a d Byn,gy n d Au,gyn d fu,gyn 1 a max d Au,fu, d fu,byn d Byn,gy n, ) ( d Au,fu ) 1/2 ( d Au,gyn 1/2 ), ( d fu,byn ) 1/2 ( d Au,gyn ) 1/ Letting,weget H Au,D 1 a d Au,z Since d z,au < H Au,D 1 a d Au,z 3.2 < d z,au, which is a contradiction, then z Au. Now, we conclude z gv Bv and z fu Au. It follows from v C g,b, u C f, A that ggv gv, ffu fu,andafu Bgv. Hence,gz z, fz z and Az Bz. Since the pair g,b is weakly compatible, gbv Bgv. Thusgz gbv Bgv Bz. Similarly, we can prove that fz Az. Consequently,z fz gz Az Bz. Therefore the maps f, g, A,andB have a common fixed point. If α intheorem 3.5, we get the following corollary.

8 8 International Mathematics and Mathematical Sciences Corollary 3.6. Let f, g : X X and A, B : X CB X satisfy H Ax,By < a d fx,gy 1 a max d Ax,fx, d By,gy, 3.21 ( d Ax,fx ) 1/2 ( d Ax,gy 1/2 ), ( d fx,by ) 1/2 ( d Ax,gy ) 1/2, for all x, y X for which the right-hand side of 3.21 is positive, where <a<1and ψ : R is a Lebesgue integrable mapping which is a summable nonnegative and such that ɛ >, R 3.22 for each ɛ>. If the following conditions (a) (d) hold: a there exists a point z f X g X which is a weak tangent point to f, g, b f, g is tangential with respect to A, B, c ffa fa, ggb gb and Afa Bgb for a C f, A and b C g,b, d the pairs f, A and g,b are weakly compatible. Then, f, g, A,andB have a common fixed point in X. If α, g f,andb A in Theorem 3.5, we get the following corollary. Corollary 3.7. Let f : X X and A : X CB X satisfy H Ax,Ay < a d fx,fy 1 a max d Ax,fx, ( d Ax,fx d Ay,fy, ) 1/2 ( d Ax,fy 1/2 ), 3.23 ( d fx,ay ) 1/2 ( d Ax,fy ) 1/2,

9 International Mathematics and Mathematical Sciences 9 for all x, y X for which the right-hand side of 3.23 is positive, where <a<1and ψ : R is a Lebesgue integrable mapping which is a summable nonnegative and such that ɛ > R 3.24 for each ɛ>. If the following conditions (a) (d) hold: a there exists sequence {x n } in X such that fx n X, b f is tangential with respect to A, c ffa fa for a C f, A, d the pair f, A is weakly compatible. Then, f and A have a common fixed point in X. If ψ t 1inTheorem 3.5, we get the following corollary. Corollary 3.8. Let f, g : X X and A, B : X CB X satisfy ( ( )) ( ) 1 αd fx,gy H Ax, By <α ( d ( Ax, fx ) d ( By, gy ) d ( Ax, gy ) d ( fx,by )) ad ( fx,gy ) { 1 a max d ( Ax, fx ),d ( By, gy ), ( d ( Ax, fx )) 1/2( ( )) 1/2, d Ax, gy 3.25 ( d ( fx,by )) 1/2 ( d ( Ax, gy )) 1/2 } for all x, y X for which the right-hand side of 3.25 is positive, where <a<1 and α. Ifthe following conditions (a) (d) holds: a there exists a point z f X g X which is a weak tangent point to f, g, b f, g is tangential with respect to A, B, c ffa fa, ggb gb and Afa Bgb for a C f, A and b C g,b, d the pairs f, A and g,b are weakly compatible. Then, f, g, A,andB have a common fixed point in X. If ψ t 1andα intheorem 3.5, we get the following corollary. Corollary 3.9. Let f, g : X X and A, B : X CB X satisfy H ( Ax, By ) <ad ( fx,gy ) { 1 a max d ( Ax, fx ),d ( By, gy ), ( d ( Ax, fx )) 1/2( ( )) 1/2, d Ax, gy ( ( )) 1/2 ( ( )) 1/2 } d fx,by d Ax, gy, 3.26

10 1 International Mathematics and Mathematical Sciences for all x, y X for which the right-hand side of 3.26 is positive, where <a<1. If the following conditions (a) (d) hold: a there exists a point z f X g X which is a weak tangent point to f, g, b f, g is tangential with respect to A, B, c ffa fa, ggb gb and Afa Bgb for a C f, A and b C g,b, d the pairs f, A and g,b are weakly compatible. Then, f, g, A,andB have a common fixed point in X. If ψ t 1, α, g f and B A in Theorem 3.5, we get the following corollary. Corollary 3.1. Let f : X X and A : X CB X satisfy H ( Ax, Ay ) <ad ( fx,fy ) { 1 a max d ( Ax, fx ),d ( Ay, fy ), ( d ( Ax, fx )) 1/2( ( )) 1/2, d Ax, fy ( d ( fx,ay )) 1/2 ( d ( Ax, fy )) 1/2 } 3.27 for all x, y X for which the right-hand side of 3.27 is positive, where <a<1. If the following conditions (a) (d) holds: a there exists sequence {x n } in X such that fx n X, b f is tangential with respect to A, c ffa fa for a C f, A, d the pairs f, A is weakly compatible. Then, f and A have a common fixed point in X. Acknowledgments W. Sintunavarat would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand SAST and the Faculty of Science, KMUTT for financial support during the preparation of this manuscript for Ph.D. Program at KMUTT. Moreover, the authors also would like to thank the National Research University Project of Thailand s Office of the Higher Education Commission for financial support under the CSEC project no Finally, the authors would like to thank Professor Frank Werner for your help and encouragement. Special thanking are also due to the reviewers, who have made a number of valuable comments and suggestions which have improved the manuscript greatly.

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